Abstract:
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations. Over the last year we identified a discrete linear system that exactly models the population of any gap across all stages of the sieve. In August 2014 we summarized our results from analyzing this discrete model on populations of single gaps. This paper extends the discrete system to model the populations of constellations of gaps. The most remarkable result is a strong Polignac result on arithmetic progressions. We had previously established that the equivalent of Polignac's conjecture holds for Eratosthenes sieve -- that every even number arises as a gap in the sieve, and its population converges toward the ratio implied by Hardy and Littlewood's Conjecture B. Extending that work to constellations, we here establish that for any even gap $g$, if $p$ is the maximum prime such that $p\# \; | g$ and $P$ is the next prime larger than $p$, then for every $2 \le j_1 < P-1$, the constellation $g,g,\ldots,g$ of length $j_1$ arises in Eratosthenes sieve. This constellation corresponds to an arithmetic progression of $j_1+1$ consecutive candidate primes.

Abstract:
We study the geometry of the Sieve of Eratosthenes. We introduce some concepts as Focals and Extremes. We find a symmetry in the distribution of the Focals (all the information about the primes is contained into a small set of numbers). We find that there is a geometric order in the Sieve and we give a formula for the greatest remainder that returns the same quotient.

Abstract:
A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations. By studying this recursion on the cycles of gaps across stages of Eratosthenes sieve, we are able to provide evidence on a number of open problems regarding gaps between prime numbers. The basic counts of short constellations in the cycles of gaps provide evidence toward the twin prime conjecture and toward resolving a series of questions posed by Erdos and Turan. The dynamic system underlying the recursion provides evidence toward Polignac's conjecture and in support of the estimates made for gaps among primes by Hardy and Littlewood in Conjecture B of their 1923 paper.

Abstract:
We establish an asymptotic formula for the number of k-difference twin primes associated with the Hawkins random sieve, which is a probabilistic model of the Eratosthenes sieve. The formula for k=1 was obtained by Wunderlich [Acta Arith. 26 (1974), 59--81]. We here extend this to k \geq 2 and generalize it to all l-tuples of Hawkins primes.

Abstract:
In this paper we consider a slightly different sieve method from Eratosthenes' to get primes. We find the periodicity and mirror symmetry of the pattern.

Abstract:
The handed down latitudinal data ascribed to Eratosthenes and Hipparchus are composed and each tested for consistency by means of adjustment theory. For detected inconsistencies new explanations are given concerning the origin of the data. Several inconsistent data can be ascribed to Strabo. Differences in Hipparchus' data can often be explained by the different types and precision of the data. Gross errors in Eratosthenes' data are explained by their origination from the lengths of sea routes. From Eratosthenes' data concerning Thule a numerical value for Eratosthenes' obliquity of the ecliptic is deduced.

Abstract:
Sieving is essential in different number theoretical algorithms. Sieving with large primes violates locality of memory access, thus degrading performance. Our suggestion on how to tackle this problem is to use cyclic data structures in combination with in-place bucket-sort. We present our results on the implementation of the sieve of Eratosthenes, using these ideas, which show that this approach is more robust and less affected by slow memory.

Abstract:
An arithmetic function $f$ is called a {\it sieve function of range} $Q$, if its Eratosthenes transform $g$ is supported in $[1,Q]\cap\N$, where $g(q)\ll_{\varepsilon} q^{\varepsilon}$ ($\forall\varepsilon>0$). Here, we study the distribution of $f$ over short {\it arithmetic bands} $\cup_{1\le a\le H}\{n\in(N,2N]: n\equiv a\, (\bmod\,q)\}$, with $H=o(N)$, and give applications to both the correlations and to the so-called weighted Selberg integrals of $f$, on which we have concentrated our recent research.

Abstract:
In this paper, we generalize the Catalan number to the $(n,k)$-th Catalan numbers and find a combinatorial description that the $(n,k)$-th Catalan numbers is equal to the number of partitions of $n(k-1)+2$ polygon by $(k+1)$-gon where all vertices of all $(k+1)$-gons lie on the vertices of $n(k-1)+2$ polygon.

Abstract:
Using a sieve procedure akin to the sieve of Eratosthenes we show how for each prime $p$ to build the corresponding M\"obius prime-function, which in the limit of infinitely large primes becomes identical to the original M\"obius function. Discussing this limit we present two simple proofs of the Prime Number Theorem. In the framework of this approach we give several proofs of the infinitude of primes.